facas9

Answered

2021-01-17

Guided Proof Let ${v}_{1},{v}_{2},....{V}_{n}$ be a basis for a vector space V.

Prove that if a linear transformation$T:V\to V$ satisfies

$T({v}_{i})=0\text{}for\text{}i=1,2,...,n,$ then T is the zero transformation.

To prove that T is the zero transformation, you need to show that$T(v)=0$ for every vector v in V.

(i) Let v be the arbitrary vector in V such that$v={c}_{1}{v}_{1}+{c}_{2}{v}_{2}+\cdots +{c}_{n}{V}_{n}$

(ii) Use the definition and properties of linear transformations to rewrite T(v) as a linear combination of$T({v}_{j})$ .

(iii) Use the fact that$T({v}_{j})=0$

to conclude that$T(v)=0,$ making T the zero transformation.

Prove that if a linear transformation

To prove that T is the zero transformation, you need to show that

(i) Let v be the arbitrary vector in V such that

(ii) Use the definition and properties of linear transformations to rewrite T(v) as a linear combination of

(iii) Use the fact that

to conclude that

Answer & Explanation

sweererlirumeX

Expert

2021-01-18Added 91 answers

a)Given:

The linear transformation

represented as

Approach:

Consider an arbitrary

The function T is said to be linear transformation if it satisfies the vector addition and scalar multiplication properties.

The linear transformation is given by,

Calculation:

As the vector set v is the subspace of V, the vector v can be written linear combination.

Write the subspace vas linear combination.

Here,

Conclusion:

Hence, it is proved above that the set

b)Given:

The linear transformation

represented as

Approach:

Consider an arbitrary

The function T is said to be linear transformation if it satisfies the vector addition and scalar multiplication properties.

The linear transformation is given by,

The vector additinon is given by,

The scalar multiplication is given by,

Calculation:

As the vector set v is the subspace of V, the vector v can be written linear combination.

Write the subspace vas linear combination.

Conclusion:

The transformation form of linear combination

c) Given:

The linear transformation

represented as

Approach:

Consider an arbitrary

The function T is said to be linear transformation if it satisfies the vector addition and scalar multiplication properties.

The linear transformation is given by,

The vector additinon is given by,

The scalar multiplication is given by,

Calculation:

Solve formula (3) with use of formula(1)

= 0

From above calculation is is clear linear transformation

satisfies

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